Top: permutation problem for two scatterers (in-vivo microcalcifications). The first singular vector of DORT (blue, circle) corresponds to the scatterer I at low frequency, and to the scatterer II at high frequency, because the reflectivity functions of the scatterers cross each other. Bottom: permutation problem in the case of 9 wires in a medical phantom (depicted in Fig. 8). One wire is significantly brighter than the other, but the other wires have similar level of reflectivity, therefore the singular values cross each other.
Unwrapping a matrix into a tensor. By remapping the indexes, the four dimensional problem is transformed into an easily solvable 2D problem.
Left: singular value decomposition for a single scatterer. Multiple nonzero singular values are presents. Middle: amplitude of the first temporal invariant. It is broadened in time because it is tuned to the most energetic frequency in order to maximize the energy for the given time window length . The second invariant (not shown) has the same envelope but a sine modulation. Right: third invariant. Its envelope has two lobes, resulting from the sum of other frequencies in the bandwidth. The fourth invariant has the same envelope as the third but a sine modulation.
The method is performed for two scatterers in an inhomogeneous medium, modeled by a near-field phase screen, with 30 ns average delay variation and 4.5 mm spatial correlation. First invariant for each of the two scatterers. The wave-fronts of the scatterers are well-separated. The invariants are tuned to different frequencies because the scatterers have different resonance frequencies (respectively, 8 and 6 Mhz). The broadening of the wave-fronts ensures that the reflected energy is the maximal for the given time window length .
Simulation of two scatterers in a homogeneous medium. Top left: typical received signal, showing the echo of the two scatterers. Top right: singular values distribution for the focused tensor; two dominant singular values are observed, they correspond to each scatterer. Bottom: the two singular matrices . They give the scatterers Green’s function. There is no broadening of the wave-fronts with this method.
Simulation of two scatterers (5 mm apart) in an inhomogeneous medium, modeled by a near-field phase screen, with 30 ns average delay variation and 4.5 mm spatial correlation. Left: typical echo showing the two wave-fronts. Middle and right: singular matrices . They correspond to the time domain Green’s function of the scatterers. The separation is not as good as in the homogeneous medium, but still decent.
Phantom used for the experiment with the region of interest in the box.
Two first singular matrices of the focused tensor. Both (left) that corresponds to the Green’s functions (wave-fronts), and that corresponds to the projection on the image pixels and therefore provides an image of the scatterers are shown. Top and bottom correspond to two different scatterers.
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